Timescales in simple models
- Exponential dacay of autocorrelations: one characteristic timescale
- Power-law decay of autocorrelations: no characteristic timescale
- Oscillations: one characteristic timescale
Noisy oscillations
Noisy oscillations contain two timescales: the oscillatory mode and the decay auf autocorrelations
Underdamped harmonic oscillator \[ \ddot{x}(t) + \zeta \omega_0 \dot{x}(t) + \omega_0^2 x^2(t) = \xi(t) \]
\[ [ x_{t+1} - 2x_t + x_{t-1} ] + \zeta\omega_0 [x_{t+1}-x_t] + \omega_0^2 x_t^2 = \xi_t \]
\[ x_{t+1} = \frac{2+\zeta\omega_0 - \omega_0^2 }{1+\zeta\omega_0} x_t - \frac{1}{1+\zeta\omega_0} x_{t-1} = \xi_t \]
The autoregressive model of order two AR(2) \[ x_{t+1} = a x_t + b x_{t-1} + \xi_t \] \[ \mbox{with} \;\; b>-1 \;\;\;\; \mbox{stationary if} \;\; 1-|a|>b \;\;\;\; \mbox{oscillates if} \;\; -a^2/4>b \]
Autocorralation function of noisy oscillation has the form \[ C(\Delta) = e^{-\Delta/\tau} \cos(\omega \Delta + \phi) \]
Example: El Nino Southern Oscillation
Superpositions
A signal composed of two independent components y,z \[ x_t = y_t + z_t \]
- The autocovariance function \[ \langle x_{t_1}x_{t_2} \rangle = \langle (y_{t_1}+z_{t_1})(y_{t_2}+z_{t_2}) \rangle = \langle y_{t_1}y_{t_2}+y_{t_1}z_{t_2}+z_{t_1}y_{t_2}+z_{t_1}z_{t_2} \rangle = \langle y_{t_1}y_{t_2} \rangle + \langle z_{t_1}z_{t_2} \rangle \] is the superposition of the autocovariance functions of both components
- The same holds for the MSD or the variance \[ \langle x^2(t) \rangle = \langle y^2(t) \rangle + \langle z^2(t) \rangle \;\;\; \mbox{or} \;\;\; \langle (x(t)-\mu_x)^2 \rangle = \langle (y(t)-\mu_y)^2 \rangle + \langle (z(t)-\mu_z)^2 \rangle \]
- So, the autocorrelation function is \[ C_{xx}(\Delta) = \frac{ \langle y_{t_1}y_{t_2} \rangle + \langle z_{t_1}z_{t_2} \rangle }{ \sigma_y^2 + \sigma_z^2 } = \frac{ \sigma_y^2 C_{yy}(\Delta) + \sigma_z^2 C_{zz}(\Delta) }{ \sigma_y^2 + \sigma_z^2 } \]
- The superposition principle also holds for the TAMSD \[ \overline{{\delta_x}_\Delta^2} = \frac{1}{t-\Delta} \sum_{n=1}^{t-\Delta} (y_{n+\Delta}+z_{n+\Delta}-y_{n}-z_{n})^2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \] \[ = \frac{1}{t-\Delta} \sum_{n=1}^{t-\Delta} (y_{n+\Delta}-y_n)^2+2(y_{n+\Delta}-y_n)(z_{n+\Delta}-z_n)+(z_{n+\Delta}-z_{n})^2 \] \[ = \frac{1}{t-\Delta} \sum_{n=1}^{t-\Delta} (y_{n+\Delta}-y_n)^2+(z_{n+\Delta}-z_{n})^2 = \overline{{\delta_y}_\Delta^2} + \overline{{\delta_z}_\Delta^2} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \]
- ... and the power spectrum \[ X(\nu) = \int \mathrm{d}t\; [y(t)+z(t)] e^{-\frac{2\pi i}{N}t\nu} = Y(\nu) + Z(\nu) \]
\[ |X(\nu)|^2 = |Y(\nu) + Z(\nu)|^2 = |Y(\nu)|^2 + |Z(\nu)|^2 \]
Measurement uncertainty and noise
No measured value is accurate - there is always some uncertainty with every value; we have to distinguish systematic errors and random errors
- Systematic errors have to be considered in models in different ways
- Example: Dynamic error for tracking particles \[ X(t)=\frac{1}{s}\int_0^sx(t-s_1)ds_1.
\]
- Uncertainty usually comes in the form of uncorrelated additive noise \[ y(t)=x(t)+\xi(t) \]
Random error - effect on statistics
- Superpostition of a white noise term and the actual signal \[ y(t)=x(t)+\xi(t) \;\;\; \mbox{with} \;\;\; \langle \xi(t)\xi(t+\Delta) \rangle = \sigma_\xi^2 \delta(\Delta) \;\;\; \mbox{and} \;\;\; \langle x(t)x(t+\Delta) \rangle = \sigma_x^2 C_{xx}(\Delta) \]
- The correlation time for a model with \[C_{xx}=e^{-\Delta/\tau}\] decreases when superimposed with white noise \[ C_{yy}(\Delta) = \int_{\epsilon\rightarrow 0}^\infty \frac{\sigma_\xi^2\delta(\Delta)+\sigma_x^2e^{-\Delta/\tau}}{\sigma_\xi^2+\sigma_x^2} \mathrm{d}\Delta = \tau \frac{\sigma_x^2}{\sigma_x^2+\sigma_\xi^2} \]
- TAMSD \[ \langle \overline{\delta^2(\Delta)} \rangle = 2\sigma_x^2 (1-e^{-\Delta/\tau})+ 2\sigma_\xi^2 \]
What is a trend?
\[ x_{t+1} = a x_t + \xi_t \;\;\;\;\;\; y_{t}=x_t+ht \]
\[ \langle \overline{\delta^2(\Delta)} \rangle = 2D\tau (1-e^{-1/\tau})+h^2t^2 \]
What is a trend? There are different interpretations
- Climate (global warming)
- Finance (inflation)
- ...
Example of Temperatures station data at Indonesia
Example of Temperatures station data at Indonesia
Summary
- Complex time series can have several characteristic time scales
- Time scales can be approximated by relaxations (overdamped motion) and noisy oscillations (underdamped motion)
- Measurement noise typically imvolves white noise, so it fluctuates on the shortest time scale
- The largest time scale in some systems is treated as a trend; it can be a 'real' (infinite) trend or just a very long timescale compared to the length of the time series